My argument for the above questions' importance:

(The following questions are primarily rhetorical, but may justify subsidiary answers)

Closing prevents answering, right? Wouldn't most of us agree that having one's question closed is a little bit discouraging at first? I acknowledge that the ideal target user will respond constructively to closing and edit the question, but let's be realistic: how often does a new user of a beta site do so? If this answer to my underlying question (which could use more answers itself!) is right in arguing that what a beta site wants most is more users, shouldn't beta moderators (and users like me with sufficient reputation to handle close votes) also be exceptionally cautious (relative to non-beta sites) about closing questions if doing so discourages a substantial proportion of users from continuing to participate? I realize that's a bigger, more amorphous question, and more likely to be a common one, so I'll ask it separately if someone searches for it and tells me that it hasn't been asked and answered yet, but I'm also not focusing on it because I see benefits of closing questions (e.g., preserves site's focus and content's quality) anyway; I'm more concerned presently with identifying the costs.

Given one assumed cost of closing questions (that it discourages some users, though it may indirectly encourage more), are there more costs? Do closed questions reduce the site's percent answered and answer ratio? My code-formatted words refer specifically to the statistics Area51.SE collects and appears to judge the health of beta sites by...am I right in reading these as focal criteria for decisions regarding the transition from beta to full launch? If so, might the costs of closing questions on a beta site outweigh the benefits for those (hypothetical people) who want to see the host site survive the beta phase above all else? Again, this question is somewhat amorphous, and really forces a difficult value choice if we're only concerned with discouraging some users (who might be less desirable users anyway) vs. maintaining site quality (and thereby encouraging other users)...but if closing questions directly impacts the statistics by which the hosting beta site's survival is decided (because closed questions can't be answered), doesn't this make voting on closing a question on a beta site much more consequential than it is on an established site?

If closed questions count against percent answered and answer ratio, then the practice of closing questions can't help increase these statistics directly, right? One can't add an answer to a closed question (most of which probably won't have any answers before getting closed), so if this inadvertently threatens the statistics by which a host site lives or dies, isn't closing a question a bit like being a sheriff who shoots himself in the foot every time he shoots a criminal?

If closed questions don't count against percent answered and answer ratio, the practice of closing questions probably will help increase these statistics, won't it? Generally, the questions one tends to close are those that haven't or probably can't be answered, so in this case, if one wants to prioritize a beta site's survival, wouldn't one want to close questions more liberally during beta if they're unlikely to receive answers? IMHO, the weeding-out process that refines all SE sites' collections of questions via feedback and democratic action is one of the primary virtues of the SE model, so if you agree, isn't this the preferable case (because it doesn't discourage this process)?

I've been active enough on threebetasites to notice that there's a recurrent theme across each: lack of consensus regarding what questions to close. This problem seems reasonable to expect of any beta site, given that another quality is lacking in beta sites by definition: definition. In these formative stages, new traffic and newly empowered users and moderators will likely clash often on issues regarding when exactly a question becomes too broad / off-topic / duplicative. One could probably find meta-questions on each of these issues on many beta sites; I've voted on some myself. Before these boundaries have been defined through experience and consensus (which are necessarily limited in the beta stage), users are more likely to act unilaterally and unchecked, which makes the balancing act of seeing that each question gets treated appropriately all the harder to maintain. It is a dynamic balance, and without consensus or an established body of sufficiently privileged users to counterbalance asymmetric activity, all it takes to push the pendulum is one user who has a rather low threshold for voting to close. If that user gets four others to agree often enough, the pendulum swings far enough to have an effect one way or another.

To focus for now on the most direct effect on beta survival via its statistics, is that effect good or bad? For now, try not to focus on indirect effects, as via site quality.

Does closing a question decrease those statistics because it sometimes prevents answering?

Answers are unlikely in most cases worth closing, but they're impossible after closing.

Or does closing a question increase those statistics because it prevents those questions from counting toward the statistics?